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arXiv · 0810.5312

Block-Structure Method for the Solution of the Matrix System of Equations g{ij}g{jk}=delta{i}{k} in the N-dimensional Case

Bogdan G. Dimitrov
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Abstract

In this paper a new block-structure method is presented for the solution of the well-known from gravity theory matrix system of equations g{ij}g{jk}=delta{i}{k} (with respect to the unknown covariant components g{ij} and by known contravariant ones g{jk}) by transforming this matrix system into a linear algebraic system of equations in the general N-dimensional case. Although powerful computer methods exist for the solution of this problem for a given (fixed) dimension of the matrices g{ij} and especially for numerical elements of g{ij}, the structure of the obtained linear algebraic system in the general N-dimensional case and for arbitrary elements of g{ij} (functions) has not been known. The proposed new analytical block-structure method for the case of symmetrical matrices g{ij} and g{jk} (the standard case in gravity theory) is based on the construction of a block-structure matrix, whose "elements" are again matrices. The method allows to obtain the structure of this linear system in the general N-dimensional case, after multiplication (to the left) with the transponed matrix. Some arguments are given why the proposed method may be applied, after some refinement and generalization for the case of non-symmetrical matrices g{ij} and g{jk}, for finding the graviton modes in the Kaluza-Klein expansion in theories with extra dimensions.

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Bogdan G. Dimitrov. 2008-10-29. Block-Structure Method for the Solution of the Matrix System of Equations g{ij}g{jk}=delta{i}{k} in the N-dimensional Case. https://arxiv.org/abs/0810.5312

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